For infinitely many solutions of ((g-1)x+10y=15) and (8x+20y=30), what is (g)?

Correct answer: (g=5)

The direct answer is option B: \(g=5\). Infinitely many solutions occur when the two equations are the same line, so the ratios of corresponding coefficients and constants must be equal: \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\). In this problem, \(\frac{10}{20}=\frac{15}{30}=\frac12\). Therefore \(\frac{g-1}{8}\) must also equal \(\frac12\). Solve step by step: \(\frac{g-1}{8}=\frac12\), so \(g-1=4\), and hence \(g=5\). Option B is correct. Option A, \(g=4\), gives \((g-1)/8=3/8\), not \(1/2\); the equations would not be identical. Option B gives \(4/8=1/2\), matching both the y-coefficient ratio and the constant ratio. Option C, \(g=6\), gives \(5/8\), so it fails the condition. Option D, \(g=7\), gives \(6/8=3/4\), also unequal to \(1/2\). It is important not to compare only one pair of coefficients: all three ratios must agree for infinitely many solutions. Geometrically, identical lines have every point of the line in common, so there are infinitely many solutions. Memory cue: first simplify the known ratios; here both are one-half, then make the unknown ratio one-half too.